Hey CW,
I do agree with all of the above. To answer one of your questions...
Yes. In general, you can say that John Lennon's songs tend to lean towards rock and Paul McCartney's towards pop. That line blurred at times, but John's poppiest songs still tend to be a lot less bubblegum than Paul's. John was more heavily influenced by 50s rock and blues-based R&B. Paul was equally influenced by those styles, but I think he enjoyed the poppier elements where John seemed to prefer the grittier, more raw sound. "Come Together" and "I Am The Walrus" where John..."Blackbird" was Paul. I think "Blackbird" worked so well because it was simple, beautiful, and it's Paul's take on the civil rights struggle. I do like Paul's songs very much, but I do have issues with some of Paul's songs being "style over substance" (especially his solo stuff from the mid-70s on...), but "Blackbird" isn't one of them...I really think that was one of his songs with a lot of feeling and sincerity.
Overall, I think the best thing about the Beatles was that they did everything and they did it perfectly...sometimes all on one album. That approach does make them very hard to pin down. There really are no albums where they flat out rock, or any that are all pop. There is a huge amount of variety on all of their albums...less pronounced on the earlier records (and maybe "Let It Be"), and extrememly pronounced on records like "Revolver" and "The White Album". Both main writers were influenced by many different styles of songwriters, and they continued to remain open to various influences as they progressed...most of which were introduced directly, or indirectly, into their songs. I really do think that their amazing scope is one of the things that made them the influential, important band that they became. The leaps between albums were unpredictable and profound...as were the leaps between songs on a single album...yet they still managed to tie those disparate styles together and make them part of a cohesive whole. They weren't following trends as much as they were setting them, and I really think that helps to make them sound fresh today while so many of their contemporaries' work sounds extremely dated.
Like I said earlier...all of the above makes it hard to label one album as the "sappiest/poppiest". I'm looking through all of my Beatles albums and I think you might enjoy a lot of the songs on the "Past Masters Volume 2" CD. It's a collection of their singles...definitely some of their best, and not many ultra poppy Paul songs at all. As far as psychedelic goes...that's easy. "Revolver" features some of the earliest psychedelic songs ever, but there manys styles on the record, including plenty of Paul's poppiest numbers. "Sgt. Pepper's" is the obvious choice, but it does tend to represent the sunnier side of psychedelia. It was mostly Paul's project. There are some great John songs too, but they are some of his poppiest and most heavily arranged (though still interesting and great). The songs may not be your favorites, but I highly recommend checking it out for the excellent production and arrangements. It affected everything that was made after it. I think of "Magical Mystery Tour" as its evil twin. The songs aren't as strong overall, but the production and arrangements are equally inventive...darker and more avant garde than Sgt. Peppers. The songs strike me as darker too...even if that's not necessarily true, subject-wise. John was doing lots of psychedelics and Paul was doing lots of cocaine, and I do think that manifests itself in both interesting and not so interesting ways. The sense of the band members going in different directions begins with this album. The so-called "White Album" may be one of the best for you to begin checking out. The mood is a little less sunny and more melancholy overall, and the stylistic range is huge. They woke from the beautiful hippie dream and there was a lot of selfish, internal tension brewing. The split between John and Pauls' writing was drastic, and they often worked on their own songs in neighboring studios at Abbey Road, simultaneously (with George or Ringo, but not each other). It includes some of Paul's most rocking songs, three excellent George Harrison songs (and one great one), and some of John's most honest songs up to this point. It also includes John's experimental tape piece, "Revolution #9", which isn't for everyone and it's certainly not sappy pop. "Let It Be" was meant to be a return to basic, traditional guitar rock so you might like that one. It includes Paul's "The Long And Winding Road", but the rest does stick to the idea of them playing as a band once again, and the songs tend to be much less experimental than the preceeding albums. It sounds like you already heard "Abbey Road". That was the last album they recorded and it is generally divided into two distinct sides...one being John's and one being Paul's (and both featuring songs by George). I think it's great, but it does lack some of the sense of inspiration and comraderie that was so appealing on their earlier albums. Still...it's amazing that they managed to forget some of their differences and create such an impressive record. I do think it sounds a little more dated than some of their others (maybe the use of the early synthesizer??), but I still enjoy it quite a bit.
I recommend trying all of their albums (especially those starting with "Beatles For Sale" and later) and see what you think. It sounds like you're bound to dislike much of Paul's output. That may be true, but I bet you'll still end up loving a minimum of 1/4 to 1/2 of every album. You can always buy them, listen to each, make your own CDs with all of your favorite songs, and re-sell the originals. Beatles CDs always sell on the used market. That's about the only way you'll ever be able to get a Beatles album with the mix if songs you prefer.
P.S. If you do find yourself preferring John's songwriting, you may want to check out his solo albums " The Plastic Ono Band" and "Imagine"...they're two of his best, and they go back to more of a basic style and leave most of the Beatles-like arrangements behind. Have fun!